On arithmetic partial differential equations

Abstract

Kovič, and implicitly Ufnarovski and Åhlander, defined a notion of arithmetic par-tial derivative. We generalize the definition for rational numbers and study several arithmetic partial differential equations of the first and second order. For some equa- tions, we give a complete solution, and for others, we extend previously known results. For example, we determine under which conditions two consecutive partial derivations are commutative.